On Strongly Quasiconvex Functions: Existence Results and Proximal Point Algorithms

Relaxed-inertial proximal point type algorithms for quasiconvex minimization

“…Moreover, we discuss different conditions which guarantee the fulfillment of the hypotheses of the convergence statements. Numerical experiments illustrate the theoretical results, as the computational results show that the relaxed-inertial proximal point type algorithm considered in this work minimizes a strongly quasiconvex function faster than the classical proximal point algorithm investigated in [21].…”

“…The above result ensures that every lower semicontinuous and strongly quasiconvex function has exactly one minimizer on every closed and convex subset K of R n . Therefore, Lemma 2.1 is useful for analyzing proximal point algorithms for classes of quasiconvex functions (see [21]).…”

“…(cf. [21,Proposition 7]) Let K ⊆ R n be a closed and convex set, h : R n → R be a proper, lower semicontinuous, strongly quasiconvex function with modulus γ ≥ 0 and such that K ⊆ dom h, β > 0 and x ∈ K. The following lemma was essentially proven in [2, Theorem 2.1] (see also [4,Lemma A.4] for a short and direct proof) and will be used in the next section to analyze the convergence of the proposed algorithm.…”

“…There are already several proximal point type algorithmic schemes for minimizing such functions in works like [12,15,20,22]. An important subclass of quasiconvex functions is the strongly quasiconvex ones (not to be confused with the semistrictly quasiconvex functions, as it sometimes happened in the literature), see e.g., [19,21,25]. It has only very recently been shown in [21] that the classical proximal point algorithm converges under standard assumptions when employed for minimizing a strongly quasiconvex function.…”

“…Motivated by works like [1,5,15,22], where inertial effects and relaxations of the iterative steps were added to classical proximal point algorithms for solving convex optimization problems in order to improve them in terms of velocity and efficiency, we augment in this paper the study from [21] by considering a relaxed-inertial proximal point type algorithm from [5] for solving optimization problems consisting in minimizing strongly quasiconvex functions whose variables lie in finitely dimensional linear subspaces. Moreover, leaving aside the inertial steps and taking convenient relaxation parameters, the constraint set can be taken only closed and convex, and the algorithm remains well-defined.…”